class: title-slide background-image: url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/USydLogo-white.svg"), url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/title-image1.jpg") background-position: 10% 90%, 100% 50% background-size: 160px, 50% 100% background-color: #0148A4 # .text-shadow[.white[STAT5003]] ## .white[Week 2: Regression and Smoothing] ### .white[Dr. Justin Wishart] ### .white[Semester 2, 2020] --- ## Readings <i class="fas fa-university fa-pull-right "></i> * Introduction to Statistical Learning James, Witten, Hastie, and Tibshirani (2013) - Chapter 3 (Linear regression) - Chpater 7.4 to 7.6 (Smoothing) --- ## Regression * Numerically fitting the model is easy .center[ <img class="image-nudge" src ="https://explainxkcd.com/wiki/images/2/24/curve_fitting.png" height=450px/> ] * Knowing how to appropriately fit the model is where you add value. --- ## The prediction problem > What is the price of a 100 sqm house in St Kilda? .center[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> ] --- ## The linear regression model `\begin{align*} Y &= \beta_0 + \beta_1 X + \varepsilon\\ & \phantom{=} \downarrow\\ y_i &= b_0 + b_1 x_i + e_i \end{align*}` * `\(X\)` is the .brand-red[predictor] (feature or independent variable) * `\(Y\)` is the .brand-red[response] (target or dependent variable) * `\(\beta_0\)` is the .brand-red[intercept] of the regression line - Expected value of `\(Y\)` when `\(X =0\)` * `\(\beta_1\)` is the .brand-red[slope] of the regression line - mean increase in `\(Y\)` for a _unit_ increase in `\(X\)` * `\(\varepsilon\)` is the .brand-red[unexplained variation] or random error. - Classically assumed to be normally distributed with mean zero and finite variance. --- ## Performance of regression estimates .columns-2[ .pull-left[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="100%" /> ] .pull-right[ * Data was simulated from model `\(Y = -0.5 + 0.75X + \varepsilon\)` * True line shown in blue * Standard linear regression fit shown in red * Why not one of the green lines? ] ] --- ## How to determine the best estimates of `\(\beta_0 + \beta_1 X\)`? * The notion of best needs a .brand-red[criterion] to measure against. .columns-2[ .pull-left[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="100%" /> ] .pull-right[ * Easiest mathematical solution is the .brand-red[least squares criterion] - Minimise the residual sum of squares `\(\displaystyle RSS = \sum_{i = 1}(y_i - \widehat y_i)^2 = \sum_{i = 1}^n e_i^2\)` ] ] --- ## Least squares equations * Can show by simple calculus the following: - Regression (slope) coefficient: `\(\displaystyle b_1 = \frac{\sum_{i = 1}^n (x_i - \overline x)(y_i - \overline y)}{\sum_{i = 1}^n (x_i - \overline x)^2} = \frac{cov(x, y)}{var(x)}\)` - Intercept: `\(b_0 = \overline y − b_1 \overline x\)` * This leads to the estimated regression line: `\begin{equation*} \widehat y = b_0 + b_1 x \end{equation*}` * Least squares regression line since it minimises the residual sum of squares. --- ## Prediction using `lm` ```r lm.fit <- lm(Price ~ BuildingArea, data = st.kilda.data) summary(lm.fit) ``` ``` ## ## Call: ## lm(formula = Price ~ BuildingArea, data = st.kilda.data) ## ## Residuals: ## Min 1Q Median 3Q Max ## -817415 -201614 -85181 19895 3403199 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -129484.0 91775.9 -1.411 0.161 ## BuildingArea 11209.5 799.8 14.015 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 490300 on 99 degrees of freedom ## Multiple R-squared: 0.6649, Adjusted R-squared: 0.6615 ## F-statistic: 196.4 on 1 and 99 DF, p-value: < 2.2e-16 ``` --- ## Standard error of population mean * Consider single population estimation problem . - Wish to estimate some mean, `\(\mu\)`, of some random variable `\(Y\)`. - If `\(Y_i\)` is sampled then `\(\widehat \mu = \overline Y\)` estimates `\(\mu\)` with $$ Var(\widehat \mu) = (SE(\widehat \mu))^2 = \frac{\sigma^2}{n} $$ - `\(\sigma^2\)` is the variance of `\(Y_i\)` - `\(n\)` is the sample size. --- ## Standard error of regression coefficient estimates * Same concept applies to the regression estimates `\begin{align*} SE(\widehat{\beta_0}) &= \sigma \sqrt{\frac{1}{n} + \frac{\overline x^2}{\sum_{i = 1}^n (x_i - \overline x)^2}}\\ SE(\widehat{\beta_1}) &= \frac{\sigma}{\sqrt{\sum_{i = 1}^n (x_i - \overline x)^2}} \end{align*}` where `\(\sigma^2 = Var(\varepsilon)\)` * As `\(n \to \infty\)`, `\(SE(\widehat \beta_0) \to 0\)` and `\(SE(\widehat \beta_1)\to 0\)` * Interestingly, if the `\(x_i\)` are more spread out, the standard errors will be smaller - more leverage to estimate the parameters. --- ## Using standard errors to compute confidence intervals ```r summary(lm.fit) # Truncated output with coefficient table ``` ``` ... ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -129484.0 91775.9 -1.411 0.161 ## BuildingArea 11209.5 799.8 14.015 <2e-16 *** ## --- ## ## Residual standard error: 490300 on 99 degrees of freedom ... ``` * We can use the standard error to estimate the 95% confidence interval as: - `\(\displaystyle (\widehat \beta_1 - t_{n - 2, 0.975}SE(\widehat \beta_1), \widehat \beta_1 + t_{n - 2, 0.975}SE(\widehat \beta_1)) = b_1 \pm t_{n - 2, 0.975} SE(b_1) = b_1 \pm t_{99, 0.975} SE(b_1)\)` * In our housing example, the 95% confidence interval for the coefficient of `BuildingArea` is [9622.6968, 12796.3032] `\begin{align*} b_1 \pm t_{n - 2, 0.975} SE(b_1) &= 1.12095\times 10^{4} \pm 1.984 \times 799.8 = (9622.6968, 12796.3032) \end{align*}` --- ## Confidence intervals of regression coefficients * More directly in <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> code. - Use the `confint` function. ```r confint(lm.fit) ``` ``` ## 2.5 % 97.5 % ## (Intercept) -311587.233 52619.18 ## BuildingArea 9622.491 12796.50 ``` * This is exact and no precision lost to rounding error. * Easy to change confidence level (99% below) ```r confint(lm.fit, level = 0.99) ``` ``` ## 0.5 % 99.5 % ## (Intercept) -370524.63 111556.57 ## BuildingArea 9108.86 13310.13 ``` --- ## Is `BuildingArea` a good predictor of price? ```r summary(lm.fit) # truncated for coefficient table ``` ``` ... ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -129484.0 91775.9 -1.411 0.161 ## BuildingArea 11209.5 799.8 14.015 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ... ``` * Linear regression assumes `\(Y = \beta_0 + \beta_1 X + \varepsilon\)` * If `BuildingArea` is not linearly related to `Price`, then `\(\beta_1= 0\)`. * Can conduct a test of significance `\(H_0: \beta_1 = 0\)` against `\(H_1: \beta_1 \neq 0\)` * Can conduct a hypothesis test by computing the `\(t\)`-statistic: $$ t = \frac{\widehat \beta_1 - \beta_1}{SE(\widehat \beta_1)} \stackrel{H_0}{=} \frac{\widehat \beta_1}{SE(\widehat \beta_1)} $$ --- ## Is `BuildingArea` a good predictor of price? ```r summary(lm.fit) # truncated for coefficient table ``` ``` ... ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -129484.0 91775.9 -1.411 0.161 ## BuildingArea 11209.5 799.8 14.015 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ... ``` * The .brand-red[p-values] for each significance test in the last column. * Recall, .brand-red[p-value] gives the probability of observing your test statistic (and other scenarios support `\(H_1\)`) assuming `\(H_0\)` is true. * Small p-value here gives very little evidence to support the claim that there is no relationship between `Price` and `BuildingArea` --- ## Goodness of fit statistic * Goodness of fit is measured by the .brand-red[coefficient of determination] or .brand-red[<i>R</i>²] `\begin{align*} R^2 &= \frac{\text{Total Sum of Squares} - \text{Residual Sum of Squares}}{\text{Total Sum of Squares}}\\ &= \frac{\sum_{i = 1}^n (y_i - \overline y)^2 - \sum_{i = 1}^n (y_i - \widehat y_i)^2}{\sum_{i = 1}^n (y_i - \overline y)^2} \end{align*}` * `\(R^2\)` is a measure between 0 and 1 * It measures the proportion of variation in the response `\(Y\)`, explained by the linear regression on `\(X\)` - A value of 0 indicates __none__ of the variance in `\(Y\)` can be explained linearly by `\(X\)` - A value of 1 indicates __all__ of the variance in `\(Y\)` can be explained linearly by `\(X\)` --- ## Linear regression fit <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> --- ## Estimating the price of a 100 `\(m^2\)` house in St Kilda ```r new.100 <- data.frame(BuildingArea = 100) predict(lm.fit, new.100, interval = "confidence") ``` ``` ## fit lwr upr ## 1 991465.5 894562.7 1088368 ``` ```r predict(lm.fit, new.100, interval = "prediction") ``` ``` ## fit lwr upr ## 1 991465.5 13820.26 1969111 ``` --- ## Fit improvements * Remove outliers: black line gives overall fit, .brand-blue[blue line] fit only to blue data (without .brand-red[red] points) <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> --- ## Linear fit after removing the outliers ```r lm.without.outliers <- lm(Price/1000 ~ BuildingArea, data = st.kilda.data, subset = BuildingArea >= 40) summary(lm.without.outliers) ``` ``` ## ## Call: ## lm(formula = Price/1000 ~ BuildingArea, data = st.kilda.data, ## subset = BuildingArea >= 40) ## ## Residuals: ## Min 1Q Median 3Q Max ## -876.75 -137.30 -18.27 109.28 896.31 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -289.254 57.471 -5.033 2.22e-06 *** ## BuildingArea 12.305 0.496 24.807 < 2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 298.5 on 97 degrees of freedom ## Multiple R-squared: 0.8638, Adjusted R-squared: 0.8624 ## F-statistic: 615.4 on 1 and 97 DF, p-value: < 2.2e-16 ``` --- ## <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> formulae * Example formula <font face="Courier New">.brand-red[Response] ~ .brand-blue[Predictor1 + Predictor2 + Predictor3]</font> * Left hand side of `~` is the .brand-red[response] variable (target to predict) * Right hand side of `~` are the the .brand-blue[predictor] variables (features) * Relationship is assumed to be additive - I.e. each additional predictor is added to explain the response `\(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \ldots\)` - Interaction or multiplicative terms are denoted with `:` and `*` which are beyond the scope fo this course. - Would be used to define other relationships - E.g. `\(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_1 X_2 + \beta_3 X_2 + \ldots\)` --- ## Multiple linear regression * Real life problems usually have more than one predictor. - Simple linear (single variable) regression can be extended to multiple predictors $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \ldots + \beta_p X_p + \varepsilon $$ * The interpretation is the `\(\beta_p\)` coefficient denotes the average increase/decrease in `\(Y\)` for each single unit increase in `\(X_p\)`, holding all the other predictors fixed. --- ## Extending the house prediction model to multiple features * Perhaps 100 `\(m^2\)` houses cost more than 100 `\(m^2\)` units? <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> --- ## Multiple regression with `lm` ```r multi.lm <- lm(Price/1000 ~ Type + BuildingArea, data = st.kilda.data) summary(multi.lm) ``` ``` ## ## Call: ## lm(formula = Price/1000 ~ Type + BuildingArea, data = st.kilda.data) ## ## Residuals: ## Min 1Q Median 3Q Max ## -700.3 -173.1 -65.9 18.6 3389.6 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 342.865 186.764 1.836 0.06945 . ## Typet -613.953 286.272 -2.145 0.03448 * ## Typeu -408.417 139.915 -2.919 0.00436 ** ## BuildingArea 9.533 1.014 9.398 2.68e-15 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 469.5 on 97 degrees of freedom ## Multiple R-squared: 0.6989, Adjusted R-squared: 0.6896 ## F-statistic: 75.06 on 3 and 97 DF, p-value: < 2.2e-16 ``` --- ## Model interpretation ```r summary(multi.lm) ``` ``` ... ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 342.865 186.764 1.836 0.06945 . ## Typet -613.953 286.272 -2.145 0.03448 * ## Typeu -408.417 139.915 -2.919 0.00436 ** ## BuildingArea 9.533 1.014 9.398 2.68e-15 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ... ``` ```r multi.pred.data <- data.frame(BuildingArea = rep(100, 3), Type = c("u", "t", "h")) predict(multi.lm, newdata = multi.pred.data) ``` ``` ## 1 2 3 ## 887.7252 682.1894 1296.1427 ``` --- class: segue-yellow background-image: url("assets/USydLogo.svg") background-size: 20% background-position: 95% 95% # Nonparametric regression or Smoothing --- ## Parametrics vs non-parametric methods * .brand-red[Parametric] methods involve selecting a statistical model (e.g. linear regression model) and fitting the parameters of the model (e.g. slope, intercept) using the training data * Nonparametric methods don't require selecting a strict model. The data is allowed to *speak for itself*. However, don’t have easily interpretable parameters. They are generally intended for .brand-red[description] rather than formal inference (e.g. `\(k\)`-nearest neighbor smoothing) --- ## Data smoothing With .brand-red[predictor-response] data, the random response variable `\(Y\)` is assumed to be a .brand-red[non-linear] function of the predictor variable `\(X\)`. $$ Y_i = f(X_i) + \varepsilon_i $$ * `\(f\)` is some fixed, non-linear smooth function. * `\(\varepsilon_i\)` is a zero-mean random variable. * Smoothing is a .brand-red[non-parametric method to estimate] `\(f\)`. --- ## Local averaging * Most smoothers (smoothing functions) rely on the concept of *local averaging* - In contrast, simple linear regression attempts to fit the best global line. * E.g. Suppose you want to determine the response `\(Y\)` conditional on `\(x\)`. - The `\(Y_i\)` whose corresponding `\(x_i\)` are near `\(x\)` should be averaged with higher weight to atttempt to estimate `\(f(x)\)`. * A generic local-averaging smoother can be written as $$ \widehat f(x) = \text{average} \left( Y_i | x_i \in N(x) \right) $$ - where average is some generalised averaging operation. - `\(N(x)\)` is some neighbourhood of `\(x\)`. --- ## Constant-Span Running Mean, `\(k\)`-nearest neighbours * A simple smoother takes the sample mean of k nearby points * We define `\(N(x_i)\)` as `\(x_i\)` itself, the `\((k-1)/2\)` points whose predictor values are nearest below `\(x_i\)`, and the `\((k-1)/2\)` points whose predictor values are nearest above `\(x_i\)` * This neighbourhood is termed the _symmetric nearest neighbourhood_, and the smoother is called a .brand-red[moving average] or a .brand-red[<i>k</i>-nearest neighbours (kNN)] smoother. * The constant-span running-mean smoother can be written as: $$ \widehat f(x_i) = mean \left[ Y_j \text{ such that} \max \left(i - \frac{k - 1}{2}, 1 \right)\le j \le \min \left(i + \frac{k - 1}{2} , n\right) \right] $$ --- ## Regression splines * Fit piece-wise functions, where each function can be a `\(d\)`-dimensional polynomial function * Constrain the function to be smooth and continuous * .brand-ref[Cubic spline] fits cubic polynomial functions, with the constraints: - continuous at each knot, continuity of the 1st derivative and continuity of the 2nd derivative * Advantage of the cubic spline is that the curve looks smooth to the eye, and can be used to fit almost any function --- ## Loess * Loess is a Locally weighted scatterplot smoothing method * The loess (__LO__cal regr__ESS__ion) smoother is a widely used method with good robustness properties. * It is essentially a weighted running-line smoother, except that each local line is fitted using a robust method rather than least squares. * As a result, the smoother is nonlinear. * Loess is computationally intensive and require densely sampled data. --- ## Local regression * Fitting local linear fits that are weighted. * Formula for local constant fit is `\(\widehat f(x) = \frac{\sum_{i = 1}^n Y_i K((X_i-x)/h)}{\sum_{i = 1}^n K((X_i-x)/h)}\)` .columns-2[ .pull-left[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="100%" /> ] .pull-right[ <img src=https://blog.motorcycle.com/wp-content/uploads/2009/05/Motorcycle-Safety-0617.jpg width='100%'/> ] ] * Sharp changes in behaviour from the acceleration on a crash test dummy. --- ## Local regression animation .columns-2[ .pull-left[ * Using a large averaging window  ] .pull-right[ * Using a smaller averaging window  ]] --- ## Nonparametric smoothing vs linear regression * Advantages of non-parametric smoothing - Can model non-linear functions (e.g. splines, loess) - Does not make any assumption about the functional form of the data * Advantages of linear regression - Computationally efficient, even for multivariate linear regression - Model is interpretable, i.e. one can know the statistical meaning of the estimated slope parameters. --- ## Reference list James, G, D. Witten, T. Hastie, et al. (2013). _An introduction to statistical learning_. Vol. 112. Springer.